Convex cones

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Nonorthogonal projection on extreme directons of convex cone

pseudo coordinates

Let LaTeX: \mathcal{K} be a full-dimensional closed pointed convex cone in finite-dimensional Euclidean space LaTeX: \mathbb{R}^n.

For any vector LaTeX: \,v\, and a point LaTeX: \,x\!\in\!\mathcal{K}, define LaTeX: \,d_v(x)\, to be the largest number LaTeX: \,t^\star such that LaTeX: \,x-t^{}v\!\in\!\mathcal{K}\,.

Suppose LaTeX: \,x\, and LaTeX: \,y\, are points in LaTeX: \,\mathcal{K}\,.

Further, suppose that LaTeX: \,d_v(x)\!=_{\!}d_v(y)\, for each and every extreme direction LaTeX: \,v_i\, of LaTeX: \,\mathcal{K}\,.

Then LaTeX: \,x\, must be equal to LaTeX: \,y\,.

proof

We construct an injectivity argument from vector LaTeX: \,x\, to the set LaTeX: \,\{t_i^\star\}\, where LaTeX: \,t_i^\star\mathrel{\stackrel{\Delta}{=}}\,d_{v_i}(x)\,.

In other words, we assert that there is no LaTeX: \,x\, except LaTeX: \,x\!=\!0\, that nulls all the LaTeX: \,t_i\,; i.e., there is no nullspace to operator LaTeX: \,d\, over all LaTeX: \,v_i\,.


Function LaTeX: \,d_v(x)\, is the optimal objective value of a (primal) conic program:

LaTeX: \,\begin{array}{cl}\mathrm{maximize}_t&t\\ \mathrm{subject~to}&x-t^{}v\in\mathcal{K}\end{array}\quad{\rm(p)}

Because dual geometry of this problem is easier to visualize, we instead interpret its dual:

LaTeX: \,\begin{array}{cl}\mathrm{minimize}_\lambda&\lambda^{\rm T}x\\ \mathrm{subject~to}&\lambda\in\mathcal{K}^*\\ &\lambda^{\rm T}v=1\end{array}~\qquad{\rm(d)}

where LaTeX: \,\mathcal{K}^*\, is the dual cone, which is full-dimensional, closed, pointed, and convex because LaTeX: \,\mathcal{K}\, is.

The primal optimal objective value equals the dual optimal value under the sufficient Slater condition, which is well known;

i.e., we assume

LaTeX: \,t^\star=\,\lambda^{\star\rm T}x\,

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